The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract.
The talks in the seminar are usually 60 minutes, including questions.

At the moment, the seminar takes place via Zoom. See schedule below for links to Zoom meetings.

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at' chalmers.se, Probability Theory), Erik Broman (broman 'at' chalmers.se, Probability Theory) or Genkai Zhang (genkai 'at' chalmers.se, Analysis).

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at' chalmers.se, Probability Theory), Erik Broman (broman 'at' chalmers.se, Probability Theory) or Genkai Zhang (genkai 'at' chalmers.se, Analysis).

**Coming seminars****2020-12-01 at 13:15**

Alexey Kuzmin (Chalmers)

Alexey Kuzmin (Chalmers)

*Reflections on the definition of the fundamental group of a C*-algebra*

Recently several approaches to the definition of the fundamental group of a C*-algebra have been made. Unfortunately they are either ad-hoc and work for a restricted class of C*-algebras, or they lack good functorial properties which one would expect to have. In this talk I will propose an alternative definition of the fundamental group influenced by the definition of the fundamental group of a scheme in algebraic geometry.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2020-12-08 at 13:15**

Marcin Lis (Vienna)

Marcin Lis (Vienna)

*The monomer-dimer model and the Neumann Gaussian Free field*

The classical dimer model is a uniform probability measure on the space of perfect matchings of a graph, i.e., sets of edges such that each vertex is incident on exactly one edge.

In two dimensions, one can define an associated height function which naturally models a ‘’uniform'' random surface (with specified boundary conditions). Moreover the model can be solved exactly which in particular means that its correlations are given by the entries of the inverse Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon who proved, already 20 years ago, that the scaling limit of the height function in bounded domains approximated by the square lattice with vanishing mesh is the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first mathematically rigorous example of conformal invariance in planar statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched. This is the so-called monomer-dimer model (or dimer model with free boundary conditions) (in our case the presence of monomers is restricted to the boundary). This modification complicates the classical analysis in several ways and I will discuss how to circumvent the arising obstacles. In the end, the main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane approximated by the square lattice with vanishing mesh is the Neumann (or free boundary conditions) Gaussian free field.

This is based on joint work with Nathanael Berestycki (Vienna) and Wei Qian (Paris).

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested in attending the talk can contact one of the organizers (please specify who you are and your reasons for interest in the talk).

**2020-12-15 at 13:15**

Siddhartha Sahi (Rutgers Univ., NJ, USA)

Siddhartha Sahi (Rutgers Univ., NJ, USA)

**2021-01-19 at 13:15**

Masatoshi Noumi (Kobe/KTH)

Masatoshi Noumi (Kobe/KTH)

**2021-01-26 at 13:15**

Jeffrey Steif (Chalmers)

Jeffrey Steif (Chalmers)

**2021-02-09 at 13:15**

Diane Holcomb (KTH)

Diane Holcomb (KTH)

**Previous seminars**

2020-10-20 at 13:15

Lyudmila Turowska (Chalmers)

*Weighted Fourier algebras and Complexification*

Fourier algebra $A(G)$ of a locally compact group $G$, introduced by Eymard, is one of the favourite objects in abstract harmonic analysis. If $G=\mathbb T$ then $A(\mathbb T)$ is the algebra of continuous function on the circle with absolutely convergent Fourier series. The algebra $A(G)$ has an advantage to be commutative that allows one to examine its Gelfand spectrum, which is known to be topologically isomorphic to $G$; the fact makes a non-trivial connection between Banach algebras and groups. We will discuss a weighted variant of Fourier algebra and show its connection with complexification of the underlying group. For compact groups this was done thanks to abstract complexification due to McKennon [Crelle, 79'] and Cartwright/McMullen [Crelle, 82']. We extended this theory to general locally compact groups and use the model to describe the Gelfand spectrum of weighted Fourier algebras, showing that the latter is a part of the complexification for a wide class of locally compact groups and weights. I shall discuss different examples of weights and determine the spectrum of the corresponding algebras, weighted Fourier algebras of $\mathbb T$, $\mathbb R$ and the Heisenberg group will be of particular interest.

This talk is based on joint work with Olof Giselsson, Mahya Ghandehari, Hun Hee Lee, Jean Ludwig and Nico Spronk. It is aimed at general mathematical audience. All notions related to Banach algebras and locally compact group theory, that are needed for the talk, will be defined.

**2020-10-27 at 13:15**

Magnus Goffeng (Lund)

Magnus Goffeng (Lund)

*Elliptic and Fredholm realizations of elliptic operators*

Boundary value problems is a well-studied topic. A few years ago Ballmann-Bär introduced a new formalism for studying boundary conditions on self-adjoint first order problems, later extended by Bandara-Bär to general first order problems. The Ballmann-Bär approach relates closely to boundary conditions posed by Atiyah-Patodi-Singer. Using work of Seeley, we extend Bandara-Bär’s work to general elliptic operators on manifolds with boundary and give a complete characterization of which boundary conditions give Fredholm realizations as well as provide a naïve index formula. Based on joint work with Lashi Bandara and Hemanth Saratchandran.

**2020-11-04 at 13:15**

Cécile Mailler (Bath)

Cécile Mailler (Bath)

*The ants walk: finding geodesics in graphs using reinforcement learning.*

Abstract: How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself? In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and this distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism. Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs, we prove that, in this model, the ants indeed eventually find the shortest path(s) between their nest and the source of food.

**2020-11-10 at 13:15**

Daniel Ahlberg (Stockholm)

Daniel Ahlberg (Stockholm)

*Title: Fixation in two-type annihilating branching random walks*

Abstract: We study a model of competition between two types evolving as branching random walks on ℤ^d. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact, in analogy to the inert chemical reaction A+B->Ø. We consider initial configurations in which the sites of ℤ^d contain one ball each, which are independently coloured red or blue according to the flip of a coin. We address the question of `fixation', referring to the sites eventually settling for a given colour, or not, and find that the answer depends on the coin determining the initial configuration being biased or not. This is joint work with Simon Griffiths and Svante Janson.

**2020-11-11 at 13:15**

Eusebio Gardella (Munster, Germany)

Eusebio Gardella (Munster, Germany)

*Title: The classification problem for free ergodic actions.*

**This seminar is cancelled.**

**2020-11-17 at 13:15**

Suresh Eswarathasan (Dalhousie, Canada)

Suresh Eswarathasan (Dalhousie, Canada)

*On the geometry of nodal domains for random eigenfunctions on compact surfaces*

A classical result of R. Courant gives an upper bound for the count of nodal domains (connected components of the complement of where a function vanishes) for Dirichlet eigenfunctions on compact planar domains. This can be generalized to Laplace-Beltrami eigenfunctions on compact surfaces without boundary. When considering random linear combinations of eigenfunctions, one can make this count more precise and pose statistical questions on the geometries appearing amongst the nodal domains: what percentage have one hole? ten holes? what percentage have their boundary being tangent 100 times to a fixed non-zero vector field? The first 20-25 minutes will give a survey on some fundamental results of Nazarov-Sodin, Sarnak-Wigman, and Gayet-Welschinger before presenting some joint works with I. Wigman (King's College London) and Matthew de Courcy-Ireland (École Polytechnique Fédérale de Lausanne) answering these questions in the last 25-30 minutes.

**2020-11-24 at 13:15**

Malin Palö (KTH)

Malin Palö (KTH)

*Wilson loop expectations in Abelian lattice gauge theories, with and without a Higgs field*

Abstract: Lattice gauge theories have since their introduction been used to predict properties of elementary particles, as approximations of Yang-Mills gauge theory. However, many of these predictions are not rigorous. Moreover, it is not clear how to obtain a limit as the lattice spacing tends to zero even in the simplest cases. A natural first step in this direction would be to attempt to understand the properties of so-called Wilson loops. In this talk, I will introduce Abelian lattice gauge theories from a probabilistic perspective and discuss some of its properties and natural observables. Also, some interesting tools which can be used to study these models will be mentioned. In particular, I will present recent results on the expected value of Wilson loops in $\mathbb{Z}_4$. This talk is based on joint work with Jonatan Lenells and Fredrik Viklund.